tactics.v

From iris.heap_lang Require Export lang.
From iris.prelude Require Import fin_maps.
Import heap_lang.

(** We define an alternative representation of expressions in which the
embedding of values and closed expressions is explicit. By reification of
expressions into this type we can implementation substitution, closedness
checking, atomic checking, and conversion into values, by computation. *)
Module W.
Inductive expr :=
  | Val (v : val)
  | ClosedExpr (e : heap_lang.expr) `{!Closed [] e}
  (* Base lambda calculus *)
  | Var (x : string)
  | Rec (f x : binder) (e : expr)
  | App (e1 e2 : expr)
  (* Base types and their operations *)
  | Lit (l : base_lit)
  | UnOp (op : un_op) (e : expr)
  | BinOp (op : bin_op) (e1 e2 : expr)
  | If (e0 e1 e2 : expr)
  (* Products *)
  | Pair (e1 e2 : expr)
  | Fst (e : expr)
  | Snd (e : expr)
  (* Sums *)
  | InjL (e : expr)
  | InjR (e : expr)
  | Case (e0 : expr) (e1 : expr) (e2 : expr)
  (* Concurrency *)
  | Fork (e : expr)
  (* Heap *)
  | Alloc (e : expr)
  | Load (e : expr)
  | Store (e1 : expr) (e2 : expr)
  | CAS (e0 : expr) (e1 : expr) (e2 : expr).

Fixpoint to_expr (e : expr) : heap_lang.expr :=
  match e with
  | Val v => of_val v
  | ClosedExpr e _ => e
  | Var x => heap_lang.Var x
  | Rec f x e => heap_lang.Rec f x (to_expr e)
  | App e1 e2 => heap_lang.App (to_expr e1) (to_expr e2)
  | Lit l => heap_lang.Lit l
  | UnOp op e => heap_lang.UnOp op (to_expr e)
  | BinOp op e1 e2 => heap_lang.BinOp op (to_expr e1) (to_expr e2)
  | If e0 e1 e2 => heap_lang.If (to_expr e0) (to_expr e1) (to_expr e2)
  | Pair e1 e2 => heap_lang.Pair (to_expr e1) (to_expr e2)
  | Fst e => heap_lang.Fst (to_expr e)
  | Snd e => heap_lang.Snd (to_expr e)
  | InjL e => heap_lang.InjL (to_expr e)
  | InjR e => heap_lang.InjR (to_expr e)
  | Case e0 e1 e2 => heap_lang.Case (to_expr e0) (to_expr e1) (to_expr e2)
  | Fork e => heap_lang.Fork (to_expr e)
  | Alloc e => heap_lang.Alloc (to_expr e)
  | Load e => heap_lang.Load (to_expr e)
  | Store e1 e2 => heap_lang.Store (to_expr e1) (to_expr e2)
  | CAS e0 e1 e2 => heap_lang.CAS (to_expr e0) (to_expr e1) (to_expr e2)
  end.

Ltac of_expr e :=
  lazymatch e with
  | heap_lang.Var ?x => constr:(Var x)
  | heap_lang.Rec ?f ?x ?e => let e := of_expr e in constr:(Rec f x e)
  | heap_lang.App ?e1 ?e2 =>
     let e1 := of_expr e1 in let e2 := of_expr e2 in constr:(App e1 e2)
  | heap_lang.Lit ?l => constr:(Lit l)
  | heap_lang.UnOp ?op ?e => let e := of_expr e in constr:(UnOp op e)
  | heap_lang.BinOp ?op ?e1 ?e2 =>
     let e1 := of_expr e1 in let e2 := of_expr e2 in constr:(BinOp op e1 e2)
  | heap_lang.If ?e0 ?e1 ?e2 =>
     let e0 := of_expr e0 in let e1 := of_expr e1 in let e2 := of_expr e2 in
     constr:(If e0 e1 e2)
  | heap_lang.Pair ?e1 ?e2 =>
     let e1 := of_expr e1 in let e2 := of_expr e2 in constr:(Pair e1 e2)
  | heap_lang.Fst ?e => let e := of_expr e in constr:(Fst e)
  | heap_lang.Snd ?e => let e := of_expr e in constr:(Snd e)
  | heap_lang.InjL ?e => let e := of_expr e in constr:(InjL e)
  | heap_lang.InjR ?e => let e := of_expr e in constr:(InjR e)
  | heap_lang.Case ?e0 ?e1 ?e2 =>
     let e0 := of_expr e0 in let e1 := of_expr e1 in let e2 := of_expr e2 in
     constr:(Case e0 e1 e2)
  | heap_lang.Fork ?e => let e := of_expr e in constr:(Fork e)
  | heap_lang.Alloc ?e => let e := of_expr e in constr:(Alloc e)
  | heap_lang.Load ?e => let e := of_expr e in constr:(Load e)
  | heap_lang.Store ?e1 ?e2 =>
     let e1 := of_expr e1 in let e2 := of_expr e2 in constr:(Store e1 e2)
  | heap_lang.CAS ?e0 ?e1 ?e2 =>
     let e0 := of_expr e0 in let e1 := of_expr e1 in let e2 := of_expr e2 in
     constr:(CAS e0 e1 e2)
  | to_expr ?e => e
  | of_val ?v => constr:(Val v)
  | _ => constr:(ltac:(
     match goal with H : Closed [] e |- _ => exact (@ClosedExpr e H) end))
  end.

Fixpoint is_closed (X : list string) (e : expr) : bool :=
  match e with
  | Val _ | ClosedExpr _ _ => true
  | Var x => bool_decide (x ∈ X)
  | Rec f x e => is_closed (f :b: x :b: X) e
  | Lit _ => true
  | UnOp _ e | Fst e | Snd e | InjL e | InjR e | Fork e | Alloc e | Load e =>
     is_closed X e
  | App e1 e2 | BinOp _ e1 e2 | Pair e1 e2 | Store e1 e2 =>
     is_closed X e1 && is_closed X e2
  | If e0 e1 e2 | Case e0 e1 e2 | CAS e0 e1 e2 =>
     is_closed X e0 && is_closed X e1 && is_closed X e2
  end.
Lemma is_closed_correct X e : is_closed X e → heap_lang.is_closed X (to_expr e).
Proof.
  revert X.
  induction e; naive_solver eauto using is_closed_of_val, is_closed_weaken_nil.
Qed.

(* We define [to_val (ClosedExpr _)] to be [None] since [ClosedExpr]
constructors are only generated for closed expressions of which we know nothing
about apart from being closed. Notice that the reverse implication of
[to_val_Some] thus does not hold. *)
Fixpoint to_val (e : expr) : option val :=
  match e with
  | Val v => Some v
  | Rec f x e =>
     if decide (is_closed (f :b: x :b: []) e) is left H
     then Some (@RecV f x (to_expr e) (is_closed_correct _ _ H)) else None
  | Lit l => Some (LitV l)
  | Pair e1 e2 => v1 ← to_val e1; v2 ← to_val e2; Some (PairV v1 v2)
  | InjL e => InjLV <$> to_val e
  | InjR e => InjRV <$> to_val e
  | _ => None
  end.
Lemma to_val_Some e v :
  to_val e = Some v → heap_lang.to_val (to_expr e) = Some v.
Proof.
  revert v. induction e; intros; simplify_option_eq; rewrite ?to_of_val; auto.
  - do 2 f_equal. apply proof_irrel.
  - exfalso. unfold Closed in *; eauto using is_closed_correct.
Qed.
Lemma to_val_is_Some e :
  is_Some (to_val e) → is_Some (heap_lang.to_val (to_expr e)).
Proof. intros [v ?]; exists v; eauto using to_val_Some. Qed.

Fixpoint subst (x : string) (es : expr) (e : expr)  : expr :=
  match e with
  | Val v => Val v
  | ClosedExpr e H => @ClosedExpr e H
  | Var y => if decide (x = y) then es else Var y
  | Rec f y e =>
     Rec f y $ if decide (BNamed x ≠ f ∧ BNamed x ≠ y) then subst x es e else e
  | App e1 e2 => App (subst x es e1) (subst x es e2)
  | Lit l => Lit l
  | UnOp op e => UnOp op (subst x es e)
  | BinOp op e1 e2 => BinOp op (subst x es e1) (subst x es e2)
  | If e0 e1 e2 => If (subst x es e0) (subst x es e1) (subst x es e2)
  | Pair e1 e2 => Pair (subst x es e1) (subst x es e2)
  | Fst e => Fst (subst x es e)
  | Snd e => Snd (subst x es e)
  | InjL e => InjL (subst x es e)
  | InjR e => InjR (subst x es e)
  | Case e0 e1 e2 => Case (subst x es e0) (subst x es e1) (subst x es e2)
  | Fork e => Fork (subst x es e)
  | Alloc e => Alloc (subst x es e)
  | Load e => Load (subst x es e)
  | Store e1 e2 => Store (subst x es e1) (subst x es e2)
  | CAS e0 e1 e2 => CAS (subst x es e0) (subst x es e1) (subst x es e2)
  end.
Lemma to_expr_subst x er e :
  to_expr (subst x er e) = heap_lang.subst x (to_expr er) (to_expr e).
Proof.
  induction e; simpl; repeat case_decide;
    f_equal; auto using is_closed_nil_subst, is_closed_of_val, eq_sym.
Qed.

Definition atomic (e : expr) :=
  match e with
  | Alloc e => bool_decide (is_Some (to_val e))
  | Load e => bool_decide (is_Some (to_val e))
  | Store e1 e2 => bool_decide (is_Some (to_val e1) ∧ is_Some (to_val e2))
  | CAS e0 e1 e2 =>
     bool_decide (is_Some (to_val e0) ∧ is_Some (to_val e1) ∧ is_Some (to_val e2))
  (* Make "skip" atomic *)
  | App (Rec _ _ (Lit _)) (Lit _) => true
  | _ => false
  end.
Lemma atomic_correct e : atomic e → heap_lang.atomic (to_expr e).
Proof.
  destruct e; simpl; repeat (case_match; try done);
    naive_solver eauto using to_val_is_Some.
Qed.
End W.

Ltac solve_closed :=
  match goal with
  | |- Closed ?X ?e =>
     let e' := W.of_expr e in change (Closed X (W.to_expr e'));
     apply W.is_closed_correct; vm_compute; exact I
  end.
Hint Extern 0 (Closed _ _) => solve_closed : typeclass_instances.

Ltac solve_to_val :=
  try match goal with
  | |- context E [language.to_val ?e] =>
     let X := context E [to_val e] in change X
  end;
  match goal with
  | |- to_val ?e = Some ?v =>
     let e' := W.of_expr e in change (to_val (W.to_expr e') = Some v);
     apply W.to_val_Some; simpl; unfold W.to_expr; reflexivity
  | |- is_Some (to_val ?e) =>
     let e' := W.of_expr e in change (is_Some (to_val (W.to_expr e')));
     apply W.to_val_is_Some, (bool_decide_unpack _); vm_compute; exact I
  end.

Ltac solve_atomic :=
  try match goal with
  | |- context E [language.atomic ?e] =>
     let X := context E [atomic e] in change X
  end;
  match goal with
  | |- atomic ?e =>
     let e' := W.of_expr e in change (atomic (W.to_expr e'));
     apply W.atomic_correct; vm_compute; exact I
  end.
Hint Extern 0 (atomic _) => solve_atomic : fsaV.
Hint Extern 0 (language.atomic _) => solve_atomic : fsaV.

(** Substitution *)
Ltac simpl_subst :=
  simpl;
  repeat match goal with
  | |- context [subst ?x ?er ?e] =>
      let er' := W.of_expr er in let e' := W.of_expr e in
      change (subst x er e) with (subst x (W.to_expr er') (W.to_expr e'));
      rewrite <-(W.to_expr_subst x); simpl (* ssr rewrite is slower *)
  end;
  unfold W.to_expr.
Arguments W.to_expr : simpl never.
Arguments subst : simpl never.

(** The tactic [inv_head_step] performs inversion on hypotheses of the
shape [head_step]. The tactic will discharge head-reductions starting
from values, and simplifies hypothesis related to conversions from and
to values, and finite map operations. This tactic is slightly ad-hoc
and tuned for proving our lifting lemmas. *)
Ltac inv_head_step :=
  repeat match goal with
  | _ => progress simplify_map_eq/= (* simplify memory stuff *)
  | H : to_val _ = Some _ |- _ => apply of_to_val in H
  | H : context [to_val (of_val _)] |- _ => rewrite to_of_val in H
  | H : head_step ?e _ _ _ _ |- _ =>
     try (is_var e; fail 1); (* inversion yields many goals if [e] is a variable
     and can thus better be avoided. *)
     inversion H; subst; clear H
  end.

(** The tactic [reshape_expr e tac] decomposes the expression [e] into an
evaluation context [K] and a subexpression [e']. It calls the tactic [tac K e']
for each possible decomposition until [tac] succeeds. *)
Ltac reshape_val e tac :=
  let rec go e :=
  match e with
  | of_val ?v => v
  | Rec ?f ?x ?e => constr:(RecV f x e)
  | Lit ?l => constr:(LitV l)
  | Pair ?e1 ?e2 =>
    let v1 := go e1 in let v2 := go e2 in constr:(PairV v1 v2)
  | InjL ?e => let v := go e in constr:(InjLV v)
  | InjR ?e => let v := go e in constr:(InjRV v)
  end in let v := go e in first [tac v | fail 2].

Ltac reshape_expr e tac :=
  let rec go K e :=
  match e with
  | _ => tac (reverse K) e
  | App ?e1 ?e2 => reshape_val e1 ltac:(fun v1 => go (AppRCtx v1 :: K) e2)
  | App ?e1 ?e2 => go (AppLCtx e2 :: K) e1
  | UnOp ?op ?e => go (UnOpCtx op :: K) e
  | BinOp ?op ?e1 ?e2 =>
     reshape_val e1 ltac:(fun v1 => go (BinOpRCtx op v1 :: K) e2)
  | BinOp ?op ?e1 ?e2 => go (BinOpLCtx op e2 :: K) e1
  | If ?e0 ?e1 ?e2 => go (IfCtx e1 e2 :: K) e0
  | Pair ?e1 ?e2 => reshape_val e1 ltac:(fun v1 => go (PairRCtx v1 :: K) e2)
  | Pair ?e1 ?e2 => go (PairLCtx e2 :: K) e1
  | Fst ?e => go (FstCtx :: K) e
  | Snd ?e => go (SndCtx :: K) e
  | InjL ?e => go (InjLCtx :: K) e
  | InjR ?e => go (InjRCtx :: K) e
  | Case ?e0 ?e1 ?e2 => go (CaseCtx e1 e2 :: K) e0
  | Alloc ?e => go (AllocCtx :: K) e
  | Load ?e => go (LoadCtx :: K) e
  | Store ?e1 ?e2 => reshape_val e1 ltac:(fun v1 => go (StoreRCtx v1 :: K) e2)
  | Store ?e1 ?e2 => go (StoreLCtx e2 :: K) e1
  | CAS ?e0 ?e1 ?e2 => reshape_val e0 ltac:(fun v0 => first
     [ reshape_val e1 ltac:(fun v1 => go (CasRCtx v0 v1 :: K) e2)
     | go (CasMCtx v0 e2 :: K) e1 ])
  | CAS ?e0 ?e1 ?e2 => go (CasLCtx e1 e2 :: K) e0
  end in go (@nil ectx_item) e.

(** The tactic [do_head_step tac] solves goals of the shape [head_reducible] and
[head_step] by performing a reduction step and uses [tac] to solve any
side-conditions generated by individual steps. *)
Tactic Notation "do_head_step" tactic3(tac) :=
  try match goal with |- head_reducible _ _ => eexists _, _, _ end;
  simpl;
  match goal with
  | |- head_step ?e1 ?σ1 ?e2 ?σ2 ?ef =>
     first [apply alloc_fresh|econstructor];
       (* solve [to_val] side-conditions *)
       first [rewrite ?to_of_val; reflexivity|simpl_subst; tac; fast_done]
  end.